Regularity of the Exercise Boundary for American Put Options on Assets with Discrete Dividends

Benjamin Jourdain (1) , Michel Vellekoop (2)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique
2 ASE - Amsterdam School of Economics

We analyze the regularity of the optimal exercise boundary for the American Put option when the underlying asset pays a discrete dividend at a known time $t_d$ during the lifetime of the option. The ex-dividend asset price process is assumed to follow Black-Scholes dynamics and the dividend amount is a deterministic function of the ex-dividend asset price just before the dividend date. The solution to the associated optimal stopping problem can be characterised in terms of an optimal exercise boundary which, in contrast to the case when there are no dividends, is no longer monotone. In this paper we prove that when the dividend function is positive and concave, then the boundary tends to $0$ as time tends to $t_d^-$ and is non-increasing in a left-hand neighbourhood of $t_d$. We also show that the exercise boundary is continuous and a high contact principle holds in such a neighbourhood when the dividend function is moreover linear in a neighbourhood of $0$.

Domaines: Finance quantitative [q-fin.CP]; Probabilités [math.PR]; Pricing [q-fin.PR]; Optimisation et contrôle [math.OC]

Benjamin Jourdain, Michel Vellekoop. Regularity of the Exercise Boundary for American Put Options on Assets with Discrete Dividends. 2009. ⟨hal-00436327v1⟩